Revisiting the dynamics of finite-sized satellite near the planet in ER3BP

Abstract

A novel approach for solving equations of motion of finite-sized satellite supposed to be moving in a proximity and around the planet in the elliptic restricted three-body problem, ER3BP is presented in this semi-analytical investigation. We consider two primaries, M Sun and m planet (the last is secondary in that binary system), both are orbiting around their barycenter on elliptic orbits. Satellite is considered to be the solid ellipsoid having nearly spherical form, with its gravitational potential to be given by a formula of MacCullagh type.Our aim is to revisit previously presented in work [Ashenberg, 1996] approach and to investigate the updated type of the satellite dynamics correlated implicitly to a kind of trapped motion (in the synodic co-rotating Cartesian coordinate system) in so way that satellite will always to be located near the secondary planet, m planet, moving on quasi-stable elliptic orbit.

On the Motion of Small Satellite near the Planet in ER3BP

“…(6-7) (z on ordinate, f on abscissa axis) for planetoid moving near the Callisto in system "Jupiter-Callisto-satellite". Also, the remarkable additional articles should be cited, which concern the problem under consideration, [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33].…”

Finite-Sized Orbiter’s Motion around the Natural Moons of Planets with Slow-Variable Eccentricity of Their Orbit in ER3BP

“…Aiming at the construction of a semi-analytical algorithm for solving, let us present Equations ( 2)-( 4) in their most obvious form for a further analysis of these equations and their solutions with regard to coordinates {x, y} (also see reference [12]):…”

“…According to the authors of references [11][12][13], let us present the system of equations in the scaled, pulsating, planar coordinate system → r = {x, y} (in an elliptical restricted three-body problem, ER3BP, at given initial conditions):…”

“….. (3) where true anomaly f is the dependent variable, f = f (t) (which is the angular distance of the radius vector from the pericenter of the orbit) determined by Equation (1), whereas r i (i = 1, 2) are the distances of the small mass m from each of the primaries with mass M planet and m moon , accordingly [12]:…”

“…One of the obvious ways [12] of semi-analytically solving the system (12) (or system (5) accomplished with Equation ( 1)) is to assume that solutions → r = {x, y} of (5) belong to the class of trapped motions of small mass m (close to the moon m moon ):…”